{"id":11324,"date":"2015-07-14T19:42:13","date_gmt":"2015-07-14T19:42:13","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=11324"},"modified":"2021-12-29T19:17:38","modified_gmt":"2021-12-29T19:17:38","slug":"solutions-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/solutions-logarithmic-functions\/","title":{"raw":"Solutions: Logarithmic Functions","rendered":"Solutions: Logarithmic Functions"},"content":{"raw":"

Solutions to Odd-Numbered Exercises<\/h2>\r\n1.\u00a0A logarithm is an exponent. Specifically, it is the exponent to which a base b<\/em>\u00a0is raised to produce a given value. In the expressions given, the base b<\/em>\u00a0has the same value. The exponent, y<\/em>, in the expression [latex]{b}^{y}[\/latex] can also be written as the logarithm, [latex]{\\mathrm{log}}_{b}x[\/latex], and the value of x<\/em>\u00a0is the result of raising b<\/em>\u00a0to the power of y<\/em>.\r\n\r\n3.\u00a0Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation [latex]{b}^{y}=x[\/latex]\\\\, and then properties of exponents can be applied to solve for x<\/em>.\r\n\r\n5.\u00a0The natural logarithm is a special case of the logarithm with base b<\/em>\u00a0in that the natural log always has base e<\/em>. Rather than notating the natural logarithm as [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex], the notation used is [latex]\\mathrm{ln}\\left(x\\right)[\/latex].\r\n\r\n7.\u00a0[latex]{a}^{c}=b[\/latex]\r\n\r\n9. [latex]{x}^{y}=64[\/latex]\r\n\r\n11.\u00a0[latex]{15}^{b}=a[\/latex]\r\n\r\n13.\u00a0[latex]{13}^{a}=142[\/latex]\r\n\r\n15.\u00a0[latex]{e}^{n}=w[\/latex]\r\n\r\n17.\u00a0[latex]{\\text{log}}_{c}\\left(k\\right)=d[\/latex]\r\n\r\n19.\u00a0[latex]{\\mathrm{log}}_{19}y=x[\/latex]\r\n\r\n21.\u00a0[latex]{\\mathrm{log}}_{n}\\left(103\\right)=4[\/latex]\r\n\r\n23.\u00a0[latex]{\\mathrm{log}}_{y}\\left(\\frac{39}{100}\\right)=x[\/latex]\r\n\r\n25.\u00a0[latex]\\text{ln}\\left(h\\right)=k[\/latex]\r\n\r\n27.\u00a0[latex]x={2}^{-3}=\\frac{1}{8}[\/latex]\r\n\r\n29.\u00a0[latex]x={3}^{3}=27[\/latex]\r\n\r\n31.\u00a0[latex]x={9}^{\\frac{1}{2}}=3[\/latex]\r\n\r\n33.\u00a0[latex]x={6}^{-3}=\\frac{1}{216}[\/latex]\r\n\r\n35.\u00a0[latex]x={e}^{2}[\/latex]\r\n\r\n37. 32\r\n\r\n39. 1.06\r\n\r\n41. 14.125\r\n\r\n43.\u00a0[latex]\\frac{1}{2}[\/latex]\r\n\r\n45.\u00a04\r\n\r\n47.\u00a0\u20133\r\n\r\n49.\u00a0\u201312\r\n\r\n51.\u00a00\r\n\r\n53.\u00a010\r\n\r\n55. 2.708\r\n\r\n57. 0.151\r\n\r\n59.\u00a0No, the function has no defined value for x\u00a0<\/em>= 0. To verify, suppose x\u00a0<\/em>= 0 is in the domain of the function [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex]. Then there is some number n<\/em>\u00a0such that [latex]n=\\mathrm{log}\\left(0\\right)[\/latex]. Rewriting as an exponential equation gives: [latex]{10}^{n}=0[\/latex], which is impossible since no such real number n<\/em>\u00a0exists. Therefore, x\u00a0<\/em>= 0 is not the domain of the function [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex].\r\n\r\n61.\u00a0Yes. Suppose there exists a real number x<\/em>\u00a0such that [latex]\\mathrm{ln}x=2[\/latex]. Rewriting as an exponential equation gives [latex]x={e}^{2}[\/latex], which is a real number. To verify, let [latex]x={e}^{2}[\/latex]. Then, by definition, [latex]\\mathrm{ln}\\left(x\\right)=\\mathrm{ln}\\left({e}^{2}\\right)=2[\/latex].\r\n\r\n63.\u00a0No; [latex]\\mathrm{ln}\\left(1\\right)=0[\/latex], so [latex]\\frac{\\mathrm{ln}\\left({e}^{1.725}\\right)}{\\mathrm{ln}\\left(1\\right)}[\/latex] is undefined.\r\n\r\n65.\u00a02","rendered":"

Solutions to Odd-Numbered Exercises<\/h2>\n

1.\u00a0A logarithm is an exponent. Specifically, it is the exponent to which a base b<\/em>\u00a0is raised to produce a given value. In the expressions given, the base b<\/em>\u00a0has the same value. The exponent, y<\/em>, in the expression [latex]{b}^{y}[\/latex] can also be written as the logarithm, [latex]{\\mathrm{log}}_{b}x[\/latex], and the value of x<\/em>\u00a0is the result of raising b<\/em>\u00a0to the power of y<\/em>.<\/p>\n

3.\u00a0Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation [latex]{b}^{y}=x[\/latex]\\\\, and then properties of exponents can be applied to solve for x<\/em>.<\/p>\n

5.\u00a0The natural logarithm is a special case of the logarithm with base b<\/em>\u00a0in that the natural log always has base e<\/em>. Rather than notating the natural logarithm as [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex], the notation used is [latex]\\mathrm{ln}\\left(x\\right)[\/latex].<\/p>\n

7.\u00a0[latex]{a}^{c}=b[\/latex]<\/p>\n

9. [latex]{x}^{y}=64[\/latex]<\/p>\n

11.\u00a0[latex]{15}^{b}=a[\/latex]<\/p>\n

13.\u00a0[latex]{13}^{a}=142[\/latex]<\/p>\n

15.\u00a0[latex]{e}^{n}=w[\/latex]<\/p>\n

17.\u00a0[latex]{\\text{log}}_{c}\\left(k\\right)=d[\/latex]<\/p>\n

19.\u00a0[latex]{\\mathrm{log}}_{19}y=x[\/latex]<\/p>\n

21.\u00a0[latex]{\\mathrm{log}}_{n}\\left(103\\right)=4[\/latex]<\/p>\n

23.\u00a0[latex]{\\mathrm{log}}_{y}\\left(\\frac{39}{100}\\right)=x[\/latex]<\/p>\n

25.\u00a0[latex]\\text{ln}\\left(h\\right)=k[\/latex]<\/p>\n

27.\u00a0[latex]x={2}^{-3}=\\frac{1}{8}[\/latex]<\/p>\n

29.\u00a0[latex]x={3}^{3}=27[\/latex]<\/p>\n

31.\u00a0[latex]x={9}^{\\frac{1}{2}}=3[\/latex]<\/p>\n

33.\u00a0[latex]x={6}^{-3}=\\frac{1}{216}[\/latex]<\/p>\n

35.\u00a0[latex]x={e}^{2}[\/latex]<\/p>\n

37. 32<\/p>\n

39. 1.06<\/p>\n

41. 14.125<\/p>\n

43.\u00a0[latex]\\frac{1}{2}[\/latex]<\/p>\n

45.\u00a04<\/p>\n

47.\u00a0\u20133<\/p>\n

49.\u00a0\u201312<\/p>\n

51.\u00a00<\/p>\n

53.\u00a010<\/p>\n

55. 2.708<\/p>\n

57. 0.151<\/p>\n

59.\u00a0No, the function has no defined value for x\u00a0<\/em>= 0. To verify, suppose x\u00a0<\/em>= 0 is in the domain of the function [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex]. Then there is some number n<\/em>\u00a0such that [latex]n=\\mathrm{log}\\left(0\\right)[\/latex]. Rewriting as an exponential equation gives: [latex]{10}^{n}=0[\/latex], which is impossible since no such real number n<\/em>\u00a0exists. Therefore, x\u00a0<\/em>= 0 is not the domain of the function [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex].<\/p>\n

61.\u00a0Yes. Suppose there exists a real number x<\/em>\u00a0such that [latex]\\mathrm{ln}x=2[\/latex]. Rewriting as an exponential equation gives [latex]x={e}^{2}[\/latex], which is a real number. To verify, let [latex]x={e}^{2}[\/latex]. Then, by definition, [latex]\\mathrm{ln}\\left(x\\right)=\\mathrm{ln}\\left({e}^{2}\\right)=2[\/latex].<\/p>\n

63.\u00a0No; [latex]\\mathrm{ln}\\left(1\\right)=0[\/latex], so [latex]\\frac{\\mathrm{ln}\\left({e}^{1.725}\\right)}{\\mathrm{ln}\\left(1\\right)}[\/latex] is undefined.<\/p>\n

65.\u00a02<\/p>\n\n\t\t\t

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